How do solve 3/(x-3)<=2/(x+2) and write the answer as a inequality and interval notation?

1 Answer
Andy Y.
Jun 13, 2017

Inequality: x <= -12 and -2 < x < 3
Interval: (-infty,-12] and (-2,3)

Explanation:

Step 1. Find the critical values by assuming equality.

Assume 3/(x-3) = 2/(x+2)

You have critical values at x=3 and x=-2 because these would cause the equation to divide by zero.

Also, solving for x gives the last critical value

3(x+2) = 2(x-3)

3x+6 = 2x-6

x = -12

Step 2. Evaluate the inequality around these critical values.

{:("Crit. Value ","Test value ", 3/(x-3) <= 2/(x+2)),(x <= -12 ," "-20," True"),(-12 < x < -2, " "-10, " False"),(-2 < x < 3," "0," True"),(x > 3, " "5," False"):}

Step 3. Complete by writing the inequality and interval notation.

Inequality: " "x <= -12 and -2 < x < 3
Interval: " "(-infty,-12] and (-2,3)

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